Strong solutions for a stochastic model of 2-D second grade fluids driven by L\'evy noise

0.21 MB·

by  Shijie Shang

Strong solutions for a stochastic model of 2-D second
grade fluids driven by L´evy noise
Shijie Shang1,, Jianliang Zhai 1,, Tusheng Zhang2,
∗ † ‡
1 School of Mathematical Sciences,
University of Science and Technology of China,
7
Hefei, 230026, China
1
0 2 School of Mathematics, University of Manchester,
2
Oxford Road, Manchester, M13 9PL, UK
n
a
J

2
Abstract: We consider a stochastic model of incompressible non-Newtonian fluids of second

R] grade on a bounded domain of R2 driven by L´evy noise. Applying the variational approach,
P global existence and uniqueness of strong probabilistic solution is established.
.
h
t
a Key Words: Second grade fluids; Non-Newtonian fluids; Stochastic partial differential
m
equations; Locally monotone condition; L´evy noise.
[

1
v 1 Introduction
4
1
3 In petroleum industry, polymer technology and problems of liquid crystals suspensions,
0
non-Newtonian fluids of differential type often arise (see [30]). It has attracted much atten-
0
. tion from a theoretical point of view, since the classical theory of Newtonian fluids is unable
1
0 to explain properties observed in the nature. In this article, we are interested in a special
7
class of non-Newtonian fluids of differential type, namely the second grade fluids, which is
1
: an admissible model of slow flow fluids such as industrial fluids, slurries, polymer melts, etc.
v
i
X Mathematically, the stochastic models for incompressible second grade fluids are de-
r scribed by the following equation:
a
d(u(t) α∆u(t))+ µ∆u(t)+curl(u(t) α∆u(t)) u(t)+ P dt
 − (cid:16)− − × ∇ (cid:17)

 = F(u(t),t)dt+σ(u(t),t)dW(t)+ f(u(t ),t,z)N(dzdt), in (0,T],

 ZZ − O× (1.1)

 divu = 0 in (0,T]; e
O×
u = 0 in ∂ [0,T];

 O ×

 u(0) = ξ in ,

 O



∗[email protected]
†[email protected]
‡[email protected]
1
where isa boundeddomainofR2 with boundary∂ of class 3,1; u = (u ,u ), P represent
1 2
O O C
the random velocity and modified pressure respectively; F(u(t),t) is the external force; W is
a one-dimensional standard Brownian motion; N is a compensated Poisson random measure.
The second grade fluids has properties of beoundedness, stability and exponential decay
(see [10]), and has interesting connections with many other fluid models, see [34, 10, 9, 11,
3, 31] and references therein. For example, as shown in [19], the second grade fluids reduces
to the Navier-Stokes equation(NSE) when α = 0, and it’s also a good approximation of the
NSE. We refer the reader to [5], [4], [24] for a comprehensive theory of the deterministic
second grade fluids.
On the other hand, in recent years, introducing a jump-type noises such as L´evy-type or
Poisson-type pertubations has become extremely popular for modeling natural phenomena,
because these noises are very nice choice to reproduce the performance of some natural
phenomena in real world models, such as some large moves and unpredictable events. There
is a large amount of literature on the existence and uniqueness solutions for stochastic
partial differential equations(SPDEs) driven by jump-type noises. We refer the reader to
[7, 1, 8, 6, 35, 37, 38, 2, 16].
We particular mention that for the stochastic 2-D second grade fluids driven by pure
jump noises, Hausenblas, Razafimandimby and Sango [17] obtained the global existence of a
martingale solution, that is the weak solution in the probabilistic sense. The purpose of this
paper is to establish the global existence and uniqueness of strong probabilistic solutions
for stochastic 2-D second grade fluids driven by general L´evy noises. To obtain our result,
we will use the so called variational approach. This approach was initiated by Pardoux [25]
and Krylov and Rozovskii [20], then further developed by many authors e.g. see [14, 13,
29, 26, 21, 2, 22]. It has been proved to be a powerful tool to establish the well-posedness
for stochastic dynamical systems with locally monotone and coercive coefficients. However,
the existing results in the literature could not cover the situation considered in this paper,
because the second grade fluids do not satisfy the coercivity condition required. Under
our consideration, we prove the main result by three steps: we first establish some non-
trivial a priori estimates of the Galerkin approximations, then we show the limit of those
approximate solutions solves the original equation by applying the monotonicity arguments,
finally we prove the uniqueness of solutions.
Finally, we mention several other results concerning the study of the stochastic models
for two-dimensional second grade fluids driven by Wiener processes. In [27, 28]. the authors
established the global existence and uniqueness of strong probabilistic solutions, and they
also investigated the long time behavior of the solution and the asymptotic behavior of the
solutions. Large deviations and moderate deviations for the stochastic models of second
grade fluids driven by Wiener processes have been established by in [39] and [40] respec-
tively. The exponential mixing property of the stochastic models of second grade fluids was
established in [36].
The organizationof thispaper is asfollows. InSection 2, we introduce some preliminaries
and state some lemmas which will be used later. In Section 3, we formulate the hypotheses
and state our main result. Section 4 is devoted to the proof of the main result.
Throughout this paper, C,C ,C ... are positive constants whose value may be different
1 2
from line to line.
2
2 Preliminaries
In this section, we will describe the framework in details, and introduce some functional
spaces and preliminaries that are needed in the paper.
Let (Ω, , ,P) be a given complete filtered probability space, satisfying the
t t∈[0,T]
F {F }
usual conditions. Let (Z, ) be a measurable space, and ν a σ-finite measure on it. For
Z
B with ν(B) < , we write
∈ Z ∞
N((0,t] B) := N((0,t] B) tν(B), t 0,
× × − ≥
for the compensated Poeisson randommeasure on[0,T] Ω Z, where N is a Poisson random
× ×
measure on[0,T] Z with intensity measure dt ν(dz). Assume that W is a one-dimensional
× ×
standard Brownian motion on the probability space (Ω, , ,P).
t t∈[0,T]
F {F }
To formulate the equation, we introduce the following spaces. For p 1 and k N,
≥ ∈
we denote by Lp( ) and Wk,p( ) the usual Lp and Sobolev spaces over respectively,
and write Hk( ) :O= Wk,2( ). LOet Wk,p( ) be the closure in Wk,p( ) of ∞O( ) the space
of infinitely diOfferentiable fOunctions w0ithOcompact supports in , aOnd denCocteOWk,2( ) by
O 0 O
Hk( ). We equip H1( ) with the scalar product
0 O 0 O
2
∂u ∂v
((u,v)) = u vdx = dx,
Z ∇ ·∇ Z ∂x ∂x
O Xi=1 O i i
where is the gradient operator. It is well known that the norm generated by this
∇ k · k
scalar product is equivalent to the usual norm of W1,2( ).
O
For any vector space X, write X = X X. Set
×
= u [ ∞( )]2 : div u = 0 ,
C ∈ Cc O
n o
V = the closure of in H1( ),
C O
H = the closure of in L2( ).
C O
We denote by ( , ) and the inner product in L2( )(in H) and the induced norm,
· · | · | O
respectively. The inner product and the norm of H1( ) are denoted respectively by (( , ))
0 O · ·
and . We endow the space V with the norm generated by the following inner product
k·k
(u,v)V := (u,v)+α((u,v)), for any u,v V,
∈
and the norm in V is denoted by V. The Poincar´e’s inequality implies that there exists a
|·|
constant > 0 such that the following inequalities holds
P
( 2 +α)−1 v 2 v 2 α−1 v 2, for any v V. (2.1)
V V
P | | ≤ k k ≤ | | ∈
We also introduce the following space
W = u V : curl(u α∆u) L2( ) ,
∈ − ∈ O
(cid:8) (cid:9)
and endow it with the semi-norm generated by the scalar product
(u,v)W := curl(u α∆u),curl(v α∆v) . (2.2)
− −
(cid:0) (cid:1)
The semi-norm in W is denoted by W. The following result states that this semi-norm
| · |
W is equivalent to the usual norm in H3( ). The proof can be found in [5, 4].
|·| O
3
Lemma 2.1 Set
W = v H3( ) : divv = 0 and v = 0 ,
∂O
∈ O |
(cid:8) (cid:9)
then the following (algebfraic and topological) identity holds:
W = W.
f
Moreover, there exists a constant C > 0 such that
v H3(O) C v W, v W. (2.3)
| | ≤ | | ∀ ∈
f
If we identify the Hilbert space V with its dual space V∗ by the Riesz representation,
then we obtain a Gelfand triple
W V W∗.
⊂ ⊂
We denote by f,v the dual relation between f W∗ and v W from now on. It is easy
h i ∈ ∈
to see
(v,w)V = v,w , v V, w W. (2.4)
h i ∀ ∈ ∀ ∈
Note that the injection of W into V is compact, thus there exists a sequence e of
i
{ }
elements of W which forms an orthonormal basis in W, and an orthogonal system in V,
moreover this sequence verifies:
(v,ei)W = λi(v,ei)V, for any v W, (2.5)
∈
where 0 < λ . Lemma 4.1 in [4] implies that
i
↑ ∞
e H4( ), i N. (2.6)
i
∈ O ∀ ∈
Consider the following “generalized Stokes equations”:
v α∆v = f in ,
− O
divv = 0 in , (2.7)
O
v = 0 on ∂ .
O
The following result can be derived from [32] and also can be found in [27, 28].
Lemma 2.2 If f V, then the system (2.7) has a unique solution v W, and the following
∈ ∈
relations hold
(v,g)V = (f,g), g V, (2.8)
∀ ∈
v W C f V. (2.9)
| | ≤ | |
Define the Stokes operator by
Au := Π∆u, u D(A) = H2( ) V, (2.10)
− ∀ ∈ O ∩
4
here the mapping Π : L2( ) H is the usual Helmholtz-Leray projection. The Helmholtz-
O −→
Leray projection implies that for any u L2( ), there exists a φ H1( ) such that
u
∈ O ∈ O
u = Πu+ φ ,
u
∇
Hence,
curl(Πv) = curl(v) v H1. (2.11)
∀ ∈
Therefore,
u W := curl(I +αA)u . (2.12)
| | | |
It follows from Lemma 2.2 that the operator (I +αA)−1 defines an isomorphism from V
into W. Moreover, for any f,g V, the following properties hold
∈
((I +αA)−1f,g)V = (f,g),
(I +αA)−1f V C(I +αA)−1f W CA f V. (2.13)
| | ≤ | ≤ | |
Let A := (I +αA)−1A , then A is a continuous linear operator from W onto itself, and
satisfies
b b
(Au,v)V = (Au,v) = ((u,v)), u W, v V, (2.14)
∀ ∈ ∈
hence b
(Au,u)V = u , u W. (2.15)
k k ∀ ∈
b
We recall the following estimates which can be found in [28].
Lemma 2.3 For any u,v,w W, we have
∈
(curl(u α∆u) v,w) C u W v V w W, (2.16)
| − × | ≤ | | | | | |
and
(curl(u α∆u) u,w) C u 2V w W. (2.17)
| − × | ≤ | | | |
Defining the bilinear operator B( , ) : W V W∗ by
· · × −→
B(u,v) := (bI +αA)−1Π curl(u α∆u) v .
− ×
(cid:0) (cid:1)
We have the following cbonsequence from the above lemma.
Lemma 2.4 For any u W and v V, it holds that
∈ ∈
B(u,v) W∗ C u W v V, (2.18)
| | ≤ | | | |
and b
B(u,u) W∗ CB u 2V. (2.19)
| | ≤ | |
In addition b
B(u,v),v = 0, u,v W, (2.20)
h i ∀ ∈
which implies b
B(u,v),w = B(u,w),v , u,v,w W. (2.21)
h i −h i ∀ ∈
b b
5
3 Hypotheses and the result
In this section, we will formulate precise assumptions on coefficients and state our main
results.
Let
F : V [0,T] V;
× −→
σ : V [0,T] V;
× −→
f : V [0,T] Z V,
× × −→
be given measurable maps. We introduce the following notations:
F(u,t) := (I +αA)−1F(u,t);
σ(u,t) := (I +αA)−1σ(u,t);
b
f(u,t,z) := (I +αA)−1f(u,t,z).
b
Applying (I+αA)−1 to thebequation (1.1), wesee that (1.1) isequivalent tothe stochastic
evolution equation
du(t)+µAu(t)dt+B u(t),u(t) dt
 (cid:0) (cid:1)
 = F(u(bt),t)+σ(ub(t),t)dW(t)+ f(u(t ),t,z)N(dzdt), (3.1)

 ZZ −
u(0) =bξ in W. b b e




Let us formulate assumptions on coefficients. Suppose that there exists constants C 0,
≥
ρ L1([0,T])andK L2([0,T])suchthatthefollowing holdforallu ,u Vandt [0,T]:
1 2
∈ ∈ ∈ ∈
(eS1) (Lipschitz continuity)
F(u ,t) F(u ,t) 2 + σ(u ,t) σ(u ,t) 2 + f(u ,t,z) f(u ,t,z) 2 ν(dz)
1 2 V 1 2 V 1 2 V
| − | | − | Z | − |
Z
ρ(t) u u 2.
1 2 V
≤ | − |
(S2) (Groewth condition)
F(u,t) 2 + σ(u,t) 2 + f(u,t,z) 2 ν(dz) K(t)+C u 2,
V V V V
| | | | Z | | ≤ | |
Z
and
f(u,t,z) 4 ν(dz) K2(t)+C u 4.
V V
Z | | ≤ | |
Z
Definition 3.1 A V-valued c`adl`ag and W-valued weakly c`adl`ag -adapted stochastic pro-
t
{F }
cess u is called a solution of the system (1.1), if the following two conditions hold:
(1) u L2(Ω [0,T];W);
∈ ×
(2) for any t [0,T], the following equation holds in W∗ P-a.s.:
∈
t t
u(t)+µ Au(s)ds+ B u(s),u(s) ds
Z Z
0 0 (cid:0) (cid:1)
t b t b t
=ξ + F(u(s),s)ds+ σ(u(s),s)dW(s)+ f(u(s ),s,z)N(dzds).
Z Z Z Z −
0 0 0 Z
b b b e
6
Now we can state the main result of this paper.
Theorem 3.2 Suppose that (S1) and (S2) are satisfied. Then for any ξ L4(Ω, ,P;W),
0
∈ F
equation (3.1) has a unique solution u. Moreover, u L4 Ω, ,P;L∞([0,T];W) .
∈ F
(cid:0) (cid:1)
4 The proof of Theorem 3.2
We divide the proof of the main result into three parts. In Section 4.1, we establish some
crucial a priori estimates for the Galerkin approximations. In Section 4.2, we prove the
existence. The uniqueness is proved in Section 4.3.
4.1 Galerkin approximations
The proof of the existence of solutions of system (3.1) is based on Galerkin approximations.
From (2.5) we know that √λ e is an orthonormal basis of V. Let V denote the n-
i i n
{ }
dimensional subspace spanned by e ,e ,......,e in W. Let Π : W∗ V be defined
1 2 n n n
{ } −→
by
n
Π g := λ g,e e , g W∗.
n i i i
h i ∀ ∈
Xi=1
Clearly, Πn V is just the orthogonal projection from V onto Vn.
|
Now, for any integer n 1, we seek a solution u to the equation
n
≥
dun(t) = Π µAun(t) B un(t),un(t) +F(un(t),t) dt+Π σ(un(t),t)dW(t)
n n
− −
 (cid:2) (cid:0) (cid:1) (cid:3)
 + Π fb(un(t ),tb,z)N(dzdt), t >b0, b (4.1)
 ZZ n −
u (0) = Π ξ, b e
n n




such that
n
un(t) = u (t)e , t 0,
jn j
≥
Xj=1
for appropriate choice of real-valued random processes u .
jn
Let
(u,t) := µAu B(u,u)+F(u,t); (4.2)
A − −
bρ(u,t) := 1+b2CBb|u|W +CA2bρ(t). (4.3)
Lemma 4.1 Under assumptions of Theorem 3.2, for alleu,u ,u W and any t [0,T],
1 2
∈ ∈
the following properties (H1), (H2) and (H3) hold:
(H1) The map s (u +su ,t),u is continuous on R.
1 2
7−→ hA i
(H2)
b
2 (u ,t) (u ,t),u u + σ(u ,t) σ(u ,t) 2 + f(u ,t,z) f(u ,t,z) 2 ν(dz)
1 2 1 2 1 2 V 1 2 V
hA −A − i | − | Z | − |
Z
ρ(ub,t) u ub 2. b b b b
2 1 2 V
≤ | − |
7
(H3)
(u,t) 2 C K(t)+ u 2 + u 4 .
W∗ V V
|A | ≤ | | | |
(cid:0) (cid:1)
If u V for some m N, thben can satisfy (H4) for any θ > 0:
m
∈ ∈ A
(H4) (Coercivity)
b
(u,t),u +θ u 2 K(t)+C u 2.
W m,θ V
hA i | | ≤ | |
Remark: since the W-norm ibs stronger than the V-norm, (H4) in general cannot be satisfied.
Proof: Since A is linear, B is bilinear, F is Lipschitz, (H1) is obvious. By (2.20) we see
that
b b b
B(u ,u ) B(u ,u ),u u
1 1 2 2 1 2
h − − i
= b B(u1 u2b,u1 u2),u1
−h − − i
= Bb(u1 u2,u1 u2),u2 .
−h − − i
So, by (2.4) (2.14), (2.13), (2.19) anbd (S1) we have
2 (u ,t) (u ,t),u u
1 2 1 2
hA −A − i
+bσ(u ,t) bσ(u ,t) 2 + f(u ,t,z) f(u ,t,z) 2 ν(dz)
1 2 V 1 2 V
| − | Z | − |
Z
b b
2bµ A(u bu ),u u 2 B(u ,u ) B(u ,u ),u u
1 2 1 2 1 1 2 2 1 2
≤− h − − i− h − − i
+ u1 bu2 2V + F(u1,t) F(u2,bt) 2V b (4.4)
| − | | − |
+ σ(u ,t) σ(ub,t) 2 + b f(u ,t,z) f(u ,t,z) 2 ν(dz)
1 2 V 1 2 V
| − | Z | − |
Z
≤|u1−b u2|2V +2bCB|u2|W|u1 −u2b|2V +CA2ρ(t)|bu1 −u2|2V
ρ(u ,t) u u 2.
≤ 2 | 1− 2|V e
This proves (H2).
By (2.19) and (S2) we get
(u,t) 2 C Au 2 + B(u,u) 2 + F(u,t) 2 C K(t)+ u 2 + u 4 ,
W∗ W∗ W∗ W∗ V V
|A | ≤ × | | | | | | ≤ | | | |
(cid:0) (cid:1) (cid:0) (cid:1)
which ibs (H3). b b b
If u Wm for some m N, there exists a constant Cm such that u W Cm u V, this
∈ ∈ | | ≤ | |
will easily yields (H4).
(cid:4)
By Lemma 4.1, according to Theorem 2 of [14], equation (4.1) has a unique global c`adla`g
solution un that satisfies the following integral equation:
t t
un(t) =Π ξ + Π (un(s),s)ds+ Π σ(un(s),s)dW(s)
n n n
Z A Z
0 0 (4.5)
t b b
+ Π f(un(s ),s,z)N(dzds),
n
Z Z −
0 Z
b e
for any t [0,T]
∈
In order to construct solutions to equation (3.1), we need to establish some a priori
estimates for un.
8
Lemma 4.2 Under assumptions of Theorem 3.2, we have
T
supE sup un(t) 4 C K2(s)ds+E ξ 4 . (4.6)
W W
n∈N (cid:16)t∈[0,T]| | (cid:17) ≤ ×(cid:16)Z0 | | (cid:17)
where K(s),s T is the function appeared in the condition (S2).
≤
Proof: For any given n N, M > 0, we define
∈
τMn := inf{t ≥ 0 : |un(t)|W ≥ M}∧T. (4.7)
Since the solution un is c`adl`ag in W and -adapted, τn is a stopping time, and moreover,
{Ft} M
τMn ↑ T P-a.s. as M → ∞. Note that |un(t)|W ≤ M, for t < τMn , so there exists a constant
C such that |un(t)|V ≤ CM whenever t < τMn .
By (2.4) and (4.1) for i = 1,...,n, we have
d(un(t),ei)V = d un(t),ei
h i
= Π (un(t),t),e dt+ Π σ(un(t),t),e dW(t)+ Π f(un(t ),t,z),e N(dzdt)
n i n i n i
h A i h i Z h − i
Z
b b e
b
=( (un(t),t),ei)Vdt+(Πnσ(un(t),t),ei)VdW(t)+ (Πnf(un(t ),t,z),ei)VN(dzdt).
A Z −
Z
b b b e (4.8)
By (2.6) and Lemma 2.2, we know that (un(t),t) W, so multiplying both sides of the
A ∈
equation (4.8) by λ , we can use (2.5) to obtain
i
b
d(un(t),ei)W = ( (un(t),t),ei)Wdt+(Πnσ(un(t),t),ei)WdW(t)+ (Πnf(un(t ),t,z),ei)WN(dzdt),
A Z −
Z
b b b e
for i = 1,...,n. Applying Itˆo formula to (un(t),e )2 and then summing over i from 1 to n
i W
yields
t t
un(t) 2W = un(0) 2W +2 ( (un(s),s),un(s))Wds+ Πnσ(un(s),s) 2Wds
| | | | Z A Z | |
0 0
t b t b
+2 (Πnσ(un(s),s),un(s))WdW(s)+2 (Πnf(un(s ),s,z),un(s ))WN(dzds)
Z Z Z − −
0 0 Z
t b b e
+ Π f(un(s ),s,z) 2 N(dzds).
n W
Z Z | − |
0 Z
b (4.9)
By (2.11) we have
curl Π curl un(s) α∆un(s) un(s) ,curl un(s) α∆un(s)
− × −
(cid:16) (cid:2) (cid:0) (cid:0) (cid:1) (cid:1)(cid:3) (cid:0) (cid:1)(cid:17) (4.10)
= curl curl un(s) α∆un(s) un(s) ,curl un(s) α∆un(s) ,
− × −
(cid:16) (cid:17)
(cid:0) (cid:0) (cid:1) (cid:1) (cid:0) (cid:1)
Noticing that (see (4.44), (4.45) and (4.46) in [27])
curl curl un(s) α∆un(s) un(s) ,curl un(s) α∆un(s) = 0, (4.11)
− × −
(cid:16) (cid:17)
(cid:0) (cid:0) (cid:1) (cid:1) (cid:0) (cid:1)
9
by (2.12), (4.2), (4.10) and (4.11) we get
( (un(s),s),un(s))W = curl (I +αA) (un(s),s) ,curl un(s) α∆un(s)
A A −
(cid:16) (cid:17)
(cid:2) (cid:3) (cid:0) (cid:1)
=(Fb(un(s),s),un(s))W +µ curl(∆un(s)),bcurl un(s) α∆un(s)
−
(cid:16) (cid:0) (cid:1)(cid:17) (4.12)
b curl Π curl un(s) α∆un(s) un(s) ,curl un(s) α∆un(s)
− − × −
(cid:16) (cid:17)
(cid:2) (cid:0) (cid:0) µ (cid:1) µ (cid:1)(cid:3) (cid:0) (cid:1)
=(F(un(s),s),un(s))W − α un(s) 2W + α curl un(s) ,curl un(s)−α∆un(s) .
(cid:16) (cid:17)
(cid:12) (cid:12) (cid:0) (cid:1) (cid:0) (cid:1)
b (cid:12) (cid:12)
Substituting the above equality into (4.9) we have
2µ t
un(t) 2 + un(s) 2 ds
W W
| | α Z | |
0
2µ t
= un(0) 2 + curl un(s) ,curl un(s) α∆un(s) ds
W
| | α Z −
0 (cid:16) (cid:0) (cid:1) (cid:0) (cid:1)(cid:17)
t t
+2 (F(un(s),s),un(s))Wds+ Πnσ(un(s),s) 2Wds
Z Z | |
0 0
t b b t
+2 (Πnσ(un(s),s),un(s))WdW(s)+2 (Πnf(un(s ),s,z),un(s ))WN(dzds)
Z Z Z − −
0 0 Z
t b b e
+ Π f(un(s ),s,z) 2 N(dzds).
n W
Z Z | − |
0 Z
b (4.13)
Applying the Itˆo formula (cf.[23]) to (4.13) we have
4µ t
un(t) 4 + un(s) 4 ds
W W
| | α Z | |
0
4µ t
= un(0) 4 + un(s) 2 curl un(s) ,curl un(s) α∆un(s) ds
W W
| | α Z0 | | (cid:16) (cid:0) (cid:1) (cid:0) − (cid:1)(cid:17)
t
+4 un(s) 2W(F(un(s),s),un(s))Wds
Z | |
0
t b t
+2 un(s) 2 Π σ(un(s),s) 2 ds+4 (Π σ(un(s),s),un(s))2 ds
W n W n W
Z | | | | Z
0 0
t b b
+4 un(s) 2W(Πnσ(un(s),s),un(s))WdW(s)
Z | |
0 (4.14)
t b
+4 un(s ) 2W(Πnf(un(s ),s,z),un(s ))WN(dzds)
Z Z | − | − −
0 Z
t b e
+4 (Π f(un(s ),s,z),un(s ))2 N(dzds)
n W
Z Z − −
0 Z
t b
+2 un(s ) 2 Π f(un(s ),s,z) 2 N(dzds)
W n W
Z Z | − | | − |
0 Z
t b
+ Π f(un(s ),s,z) 4 N(dzds)
n W
Z Z | − |
0 Z
t b
+4 (Πnf(un(s ),s,z),un(s ))W Πnf(un(s ),s,z) 2WN(dzds).
Z Z − − | − |
0 Z
b b
10